Beliefs, Market Size and Consumer Search

Transcription

1 Beliefs, Market Size and Consumer Search (PRELIMINARY AND INCOMPLETE) Maarten Janssen and Sandro Shelegia February 15, 2014 Abstract We analyze two unexplored aspects of the Wolinsky model (Wolinsky (1986) is commonly adopted to understand the implications of sequential consumer search with differentiated goods): (i) the equilibrium price depends on consumers out-of-equilibrium beliefs, and for some beliefs the price may be independent of firms marginal cost; (ii) demand drops drastically when prices exceed reservation utility. We then use these two observations in a model where consumers search for better prices and product matches, and are uninformed about the retailers cost. Uncertainty about retailers cost may be twofold. First, when retailers common costs follow a random distribution, the realization of which is known to the retailers, but not to consumers, we show that retail prices are higher than in the standard setting where consumers observe the retail cost. Moreover, prices are random (responding to random cost shocks) but there may be a regular price that firms charge for a continuum of cost realizations. Second, when retailers cost is determined by an upstream monopolist, prices are initially increasing and then decreasing in search cost, and there may be discontinuous jumps in prices. Other aspects of the Wolinsky model are analyzed. JEL Classification: D40; D83; L13 Keywords: Vertical Relations, Consumer Search, Double Marginalization, Product Differentiation. 1 Introduction The consumer search literature analyzes how markets function when consumers have to spend resources on acquiring information concerning prices and how well products fit their tastes. The equilibrium analysis has firms setting prices and consumers deciding on their optimal search strategy. To determine optimal consumer search strategies it is important to specify what price-product combination consumers believe they will encounter in case they would continue to search. If consumers observe offers that may occur along the equilibrium path, then these believes are specified by the equilibrium strategies of the firms. However, and this may often be important for the equilibrium characterization itself, the beliefs after off-the-equilibrium path offers are not restricted by the equilibrium definition. By and large the literature has implicitly or explicitly assumed that consumers hold passive beliefs, i.e., after observing a deviation from the equilibrium path, consumers still believe that other firms stick to their equilibrium strategies. In principle, however, there is no reason to ignore alternative beliefs, for example that consumers hold symmetric beliefs, i.e., after observing a deviation from the equilibrium path, consumers believe Department of Economics, University of Vienna. maarten.janssen@univie.ac.at Department of Economics, University of Vienna. sandro.shelegia@univie.ac.at 1

2 that other firms have also deviated in the same way. In this paper we show that these symmetric beliefs are natural in two settings where firms (retailers ) common cost is either determined exogeneously in a random fashion that is not observed by consumers, or determined endogenously by an upstream firm. We show that these symmetric beliefs give rise to results that are both qualitatively and quantitatively different from the predictions of the traditional model with passive beliefs. We take the seminal model of Wolinsky (1986) where cost is known and consumers have passive beliefs as our starting point. Most papers that build on this model (see, e.g., Anderson and Renault (1999), Anderson and Renault (2006), Armstrong et al. (2009), Bar-Isaac et al. (2011)) analyze the situation where search costs are relatively small and consumers reservation utility is not binding for the firms optimization problem. The behaviour of the model for larger search cost is relatively under-explored. It turns out that when the search cost is such that consumers reservation utility becomes binding, firm set the monopoly price (that is independent of the search cost), and price is continuous in search cost and weakly increasing. More importantly, equilibrium market demand drops discretely at the critical level of search cost. The reason is that, after having observed that the product match with the first visited firm is not satisfactory, consumers do not continue to search and therefore will not find out whether the product of the second firm better matches with their preferences. This market size effect is the second main ingredient of our results in the setting with vertical market structure. With symmetric beliefs in the Wolinsky model, it is not difficult to see that firms will set higher prices. This is becuase consumers are less willing to continue to search after observing a deviation to a higher price with symmetric beliefs than with passive belief (with symmetric beliefs they think that the other firm has also increased its price) than the equilibrium price under passive beliefs. This price increasing effect drives our first main result, namely that under symmetric beliefs, prices are non-monotonic in the search cost. The reason can be understood as follows. For large search cost, as consumer behavior is not driven by their beliefs about prices at the other firm, price remains equal to the monopoly price. For very small search cost, the equilibrium price remains smaller than consumers reservation utility and here prices are increasing in search cost and larger than under passive beliefs. As the reservation utility is independent of whether consumers have passive or symmetric beliefs, this utility becomes binding at lower levels of the search cost with symmetric beliefs. At these intermediate levels of search cost where the reservation utility is binding for symmetric but not for passive beliefs, equilibrium prices are equal to the reservation utility becuase. As this utility is decreasing in search cost, so are equilibrium prices. These observations are of importance for understanding the implications of asymmetric information about common cost components between retailers and consumers. Symmetric beliefs are natural in such a setting where retailers common cost is exogenously determined and follows a random process. After observing a retail price by one firm, consumers form beliefs about the cost that is common to all retailers. Along the equilibrium path, upon observing a high price, consumers will infer that common cost is high and that the other retailer will also set the same price. Given our result on the Wolinsky model with symmetric beliefs, it is not surprising that in this setting with small search cost there exists a symmetric equilibrium where for each cost realization prices (and firms profits) are strictly larger than in the corresponding model where cost is known to consumers and they have passive beliefs. Interestingly, the equilibrium price can be as high as the joint profit maximizing price. 1 When this happens, the joint profit maximizing price equals consumers reservation utility. It also follows that for a given cost realization, retail price 1 The joint profit maximizing price is the price that would be set by a m monopolist that sells both (differentiated) products. 2

3 may be decreasing in search cost when search cost is at intermediate values. An interesting feature that plays an important role in the vertical relations model discussed below is that for most values of the search cost, the equilibrium pricing strategy has a flat part where price does not vary with cost, apart from parts where price is increasing in cost. Thus, for many parameter values the equilibrium is neither pooling, nor separating, but semiseparating. The reason is that there is a region of cost levels where the equilibrium price equals consumers reservation utility and this reservation utility is independent of cost. We then turn to the results of our paper dealing with vertical relations, where the retailers common cost is set by an upstream firm. In this setting, after observing a price off the equilibrium path, consumers have to think about whether it is the retailer or the manufacturer that has deviated. 2 We show that in this vertical relations model with search, a combination of interesting features that have been discussed above in isolation can arise. To do so, we add to the Wolinsky model an upstream firm setting the wholesale price that becomes retailers marginal cost. In the benchmark vertical model consumers observe this wholesale price. We then compare the equilibrium of that model with the case where the wholesale price remains unknown to consumers. When wholesale price is observed, but search cost is small, we have a model where the upstream firm chooses its profit-maximizing price given the margins that downstream firms will charge. For small search costs downstream prices are smaller than the consumers reservation utility and increasing in search cost, even though upstream prices are falling. Once the search cost crosses a critical level, the retail price becomes larger than the consumers reservation price (with the corresponding drop in market demand as a consequence) in case the manufacturer does not drastically adjust its behavior. As the manufacturer has a clear incentive that consumers who are not satisfied at the first retailer continue to search to see whether the product of the second retailer better fits their taste, it will prevent the market demand to drop by lowering its price significantly. This leads to retail prices also falling in search cost. When search cost rises further there is a point where the manufacturer has to set such a low price to prevent the market demand from falling that it is no longer in its interest to do so. At this point, wholesale and retail prices drastically increase to the classic double marginalization level. Consumers stop searching and for all larger search cost the equilibrium prices are constant. When retail costs are not observed by consumers, there are multiple equilibria depending on to whom consumers attribute a deviation from the expected (equilibrium) retail price. As retail prices are now the outcome of the combination of a retail pricing strategy and the actual upstream price chosen, a consumer cannot know whether the upstream firm or the retailer has deviated after observing a deviation and both passive and symmetric beliefs can be rationalised. Under passive beliefs, the equilibrium structure is very similar to the case where wholesale price is observed. The only difference is that the manufacturer may have incentive to charge a slightly different price due to the fact that its deviations are not observed. With symmetric beliefs, for a given wholesale price downstream behavior is exactly as in the random cost model. This fact creates interesting incentives for the manufacturer. Namely, at a certain level of search cost, the equilibrium wholesale price increases discontinuously as downstream equilibrium prices are independent of retail cost. Other features related to the market size effect discussed for the case where the wholesale price is observed also do hold here. Thus, in the case where the wholesale arrangement is unobserved by consumers, wholesale prices can be increasing in search cost, decreasing in search cost, and may perform discontinuous jumps. Retail prices fall with search costs in a surprisingly large interval 2 Consumers can, of course, blame the manufacturer in case they do not observe the contractual arrangements between manufacturer and retailers and therefore do not observe the wholesale price (which is the retail cost). In the context of consumer search where consumers are not supposed to know the retail price, this is probably the most natural setting to consider. 3

4 and the quantitive effects we show in numerical examples can be quite substantial. Thus, the forces created by symmetric beliefs and the market size effect can be empirically important. This paper is related to several strands in of literature. In the context of the homogeneous goods search model by Stahl (1989), Dana (1994), and more recently, Tappata (2009), Chandra and Tappata (2011) and Janssen et al. (2011) have analyzed an asymmetric information model where firms know their common cost and consumers do not. 3 These papers show that for the same average cost realization equilibrium prices in the asymmetric information model are higher than in the setting where the cost is known to be equal to the average cost of the cost uncertainty model. Our results in this paper for the asymmetric information model are stronger in that we show that for all cost realizations prices are higher. Moreover, in our asymmetric information model prices are non-monotonic in search cost, whereas they are monotonically increasing in the homogeneous goods search models. The difference in results is related to the difference in underlying mechanism: in our paper the main mechanism builds on symmetric beliefs and the market size effect, while symmetric beliefs would destroy the equilibrium analysis in the homogeneous goods search literature. The importance of beliefs in a vertical structure has recently also been stressed by Pagnozzi and Picolo (2012). In that paper, the authors study the reason for manufacturers to sell to consumers via an independent retailer rather than directly. In their framework, it is retailers that have to form beliefs about the wholesale prices that manufacturers charge to their competitors. They show that the nature of the vertical channel depends on whether retailers hold passive or symmetric beliefs. Janssen and Shelegia (2013) perform the vertical analysis of the Stahl (1989) model where retailers cost is determined by a manufacturer maximizing profits. They consider two different scenarios, one where consumers do not observe retailers cost and one where they do. they show that the unobservaibility of retailers cost has severe implications for the prices charged in the market due to the fact that market demand is much more inelastic when the wholesale price is unobserved. The analysis of the vertical structure in the current paper mainly relies on mechanisms (symmetric beliefs and the market size effect) that are absent in the early paper. The issues we touch upon in this paper also have some similiarity to the issues recently discussed in the literature on recommended retail prices (see, e.g., Lubensky (2011) and Buehler and Gärtner (2012)). Lubensky suggests that by recommending retail prices manufacturers provide information to consumers on what reasonable retail prices to expect in an environment where the manufacturer s marginal cost is random and only known to the manufacturer. Manufacturer thereby affects consumers search behaviour. In our framework, when informed about the wholesale price, consumers have a better notion of how large retail margins. In Buehler and Gärtner (2012) consumers are not strategic and the recommended retail price is used by the manufacturer to communicate demand and cost information to the retailer. The rest of the paper is organized as follows. The next section sets out the model we use to analyze the pricing implications of the different environments we study. Section 3 then analyses the implications of symmetric beliefs in the Wolinsky model and it shows the under-explored market size effect. Section 4 discusses the implications of asymmetric information concerning retailers common cost. Interestingly, the increased price effect does not disappear if cost uncertainty vanishes. Section 5 then analyzes the effects of asymmetric information concerning retailers common cost in an environment where this cost is not a random variable, but strategically chosen by an upstream firm. The last 3 In the context of different models, Benabou and Gertner (1993) and Fishman (1996) have also studied the effects of asymmetric information about retailers cost in consumer search models. 4

5 section concludes. 2 The model The retail side of the three models is exactly the same as in Wolinsky (1986). There are two retail firms, 1 and 2, who can acquire some input at a common cost c. The retailers transform the input costlessly into a final differentiated good, using a one for one technology. There is a unit mass of consumers per firm. Utility to a consumer from buying the good at firm i is v i. This utility is drawn according to some distribution function G(v i ), which is defined over the interval [v, v]. The valuation of a consumer for the product of firm 1 is independent of his valuation for the product of firm 2. Each consumer costlessly visits one of the downstream firms at random and finds out v i and p i. After visiting the first firm, consumers have to decide whether to buy there, search the second retail firm, or to stop searching altogether. The additional search comes at a cost s. The consumer can always go back to the first firm at no additional cost (free recall). In the search literature based on the Wolinsky model, it is common to denote by w the expected utility of searching another firm including the search cost. Consumers visit the first firm for free, 4 and after observing the match value v i and the price p i they decide whether or not to visit the second firm. If they do so, they make their purchase at the best available surplus v i p i. If they decide not to continue searching, they buy at the first firm if v i p i 0. In general, an individual retailer i s demand depends on the price p i it chooses, the price p j of firm j and on consumers beliefs p e j about the price of firm j. The latter may depend on the marginal cost c and price charged by firm i. For now we will keep the formulation general, and specify model-specific beliefs later. Consider a consumer who visits firm i. To determine the reservation utility w we have to determine the benefit from searching firm j. Under free recall, the reservation utility w if prices are equal at both firms, is the solution to: v (v w )f(v) dv = s. w If such a solution does not exist, then w = v. This means that a consumer who draws v i and p i will search the other firm if v i < w p e j + p i. For consumers to ever continue to search we need that w > p e j. If this inequality is not satisfied, a consumer will not continue to search. Given the optimal search behaviour from above, firm 1 s demand is given by Q 1 (p 1 ) = (1 G(w p e 2 + p 1 )) + G(w p e 1 + p 2 )(1 G(w p e 1 + p 1 )) (1) + w p e 2 +p 1 p 1 G(p 2 p 1 + v)g(v)dv + w p e 1 +p 1 p 1 G(p 2 p 1 + v)g(v)dv. The first term is the demand from consumers who visit firm 1 and buy outright because their v 1 falls below the threshold w p e 2 + p 1. The second term is the demand from consumers who visit firm 2, draw v 2 lower than the threshold w p e 1 + p 2 move on to firm 1, encounter there v 1 more than w p e 1 + p 1 and buy at firm 1. The first integral is the demand from those who move on from firm 1 to firm 2, but come back to firm 1. Finally, the second integral is the demand from those consumers who first visit firm 2, move on to firm 1 and buy there. 4 Here, we follow most of the consumer search literature. When desired we show, however, how our results depend on this assumption. 5

6 To understand the role of beliefs in the Wolinsky model, we formally define the equilibrium notion below. As, depending on the out-of-equilibrium beliefs, consumers beliefs about the next price to be observed may depend on the price p 1 observed in the first search, consumers expected utility (or reservation utility) of searching another firm may depend on the price observed at the first firm, and we write w (p 1 ). Definition 1 A symmetric perfect Bayesian equilibrium of the Wolinsky model is a price p (the same for both firms) and a reservation utility w (p) such that 1. Each firm i chooses p i = p to maximize its expected profit given the price of the other firm and consumers reservation utility; 5 2. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 3. Consumers beliefs are updated using the Bayes Rule when possible, i.e., whenever they observe p 1 = p at their first search, they believe that the other firm has also set a price p. Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are either i. (passive beliefs) such that p e 2 = p if p 1 p, or ii. (symmetric beliefs) such that p e 2 = p 1 if p 1 p. In principle, one could study the implications of many different formulations of outof-equilibrium beliefs. For example, one could specify a convex combination of passive and symmetric beliefs or one could make the belief about the price the other firm has set conditional on the price and have a different belief for any out-of-equilibrium price. Passive beliefs is the common assumption in the search literature and we want to study the impact of that assumption by studying the implications of another belief formation process that is considered in another literature (see, e.g., Hart and Tirole (1990) and McAfee and Schwartz (1994) in the context of vertical relations between a manufacturer and two retailers what we also consider later). As explained in the Introduction, we also (and mainly) want to supplement the Wolinsky model with settings where the firms have a common marginal cost. First, we consider a market where the common cost c follows a random distribution with support [c, c]. Firms know the realization of c, but consumers do not so that we have asymmetric information between firms and consumers. One can think of this as that the common cost is determined in a competitive (upstream) market environment where price is determined by supply and demand. Consumers think it is highly unlikely (a probability 0 event) that cost is below c or above c. A second environment we consider is where the common cost is determined by an upstream monopolist. The upstream firm U produces the essential input at a marginal cost of zero and charges c for one unit to the downstream competitors. In this second environment we study a benchmark model where retailers cost is observed by consumers and a a model where it is not observed. The timing of these two games is that the common cost is determined either by Nature or by an upstream firm. The choice of c is observed by the downstream firms (and, in the benchmark vertical relations model, but only there, also by consumers). After c is determined, we have the same interaction between retailers and consumers as described above. In the asymmetric information model where retailers cost is randomly determined, retailers (symmetric) strategy is given by a function p(c), whereas the search strategy is characterized w (p 1 ). A symmetric equilibrium for this model is then given as follows. 5 Firm i will set a price that solves the first order condition p 1 = c Q 1, where Q 1 (p Q1 is given in (1). 1) 6

7 Definition 2 A symmetric perfect Bayesian equilibrium in continuous strategies of the search model with asymmetric information about cost is a continuous pricing strategy p (c) with reach P and a reservation utility w (p 1 ) such that 1. for every cost realization c, each firm i chooses p i (c) = p (c) to maximize its expected profit given the pricing strategy p (c) of the other firm and consumers reservation utility w (p 1 ); 2. consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 3. consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 P at their first search, they believe that the other firm has also set a price p e 2 = p 1. Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are symmetric, i.e., p e 2 = p 1 if p 1 / P. Note that in the asymmetric information model, for any equilibrium price p i P, consumers have to have beliefs that satisfy p e j = p i. The reason is simple: if equilibrium is symmetric, and consumers observe one (or possible many) equilibrium prices, they have no other choice but to believe that the firm they visited first is playing the equilibrium, and thus the other firm is also playing the same equilibrium and charges the same price. Again, there are in principle no restrictions on beliefs after observing out-of-equilibrium prices. The problem with passive beliefs is, however, that it is typically not clear what passive beliefs mean in this context. In equilibrium, all prices in P may be observed and if another price is observed upon the first search, then passive beliefs are only meaningful in a pooling equilibrium where P is a singleton. We therefore assume that out-of-equilibrium beliefs are similar to equilibrium beliefs and are symmetric: p e j = p i. In the model where retailers cost are set by an upstream firm and the decision of this upstream firm is not observed by the consumer, a symmetric equilibrium is defined in a similar way. Of course, the equilibrium for this setting has to introduce the manufacturer as a separate player. In addition, we can have both passive and symmetric out-of-equilibrium beliefs as consumers may either blame the retailer that was visited for having deviated (assuming the wholesale price is at its equilibrium level) resulting in passive beliefs, or they may blame the manufacturer (assuming the retailers choose their equilibrium strategies and simply react to a chance in the wholesale price) resulting in symmetric beliefs. Definition 3 A symmetric perfect Bayesian equilibrium of the search model with vertical relations is a wholesale price c, a retail pricing strategy p (c) and a reservation utility w (p 1 ) such that 1. the manufacturer chooses c so as to maximize its profit given the pricing strategy p (c) of the retailers and consumers reservation utility w (p); 2. each retailer i chooses p i (c) = p (c) to maximize its expected profit given the pricing strategy p (c) of the other retailer and consumers reservation utility w (p); 3. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 4. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p i = p (c ) at their first search, they believe that the other firm has also set a price p e j = p (c ). Out-of-equilibrium beliefs p e j of the price set by the firm that is not yet visited are either 7

8 i. (passive beliefs) such that p e j = p (c ) if p i p (c ), or ii. (symmetric beliefs) such that p e j = p i if p i p (c ). When, as in the benchmark vertical relations model, the consumer observes the common retailer s cost, the consumer search strategy can depend on this cost and the expected utility of searching another firm is denoted by w (p, c). Other aspects of the equilibrium definition remain the same. 3 The Wolinsky Model The reference point for the environments we consider in this paper is the Wolinsky model where two retailers have the same cost c. In this section, we reconsider the Wolinsky model, show how under passive beliefs the market size effect comes about and then analyze the implications of symmetric beliefs. In order to find the symmetric equilibrium price p for the Wolinsky model under passive beliefs, we use (1) and adopt it to the Wolinsky model. In particular, we then have that firm 2 charges the equilibrium price, and consumers who visit firm 1 or firm 2 first also expect the other firm to charge the equilibrium price no matter what the price is that the consumer observes, so p 2 = p e 2 = pe 1 = p. When the equilibrium price satisfies w p, expected demand for firm 1 then simplifies to: Q 1 = (1 G(w p + p 1 ))(1 + G(w )) + 2 w p +p 1 p 1 G(p p 1 + v)g(v) dv. The first order condition for firm 1 s profit maximization, along with the equilibrium condition p 1 = p gives then the following price setting rule for both retailers: p 1 G(p ) 2 (c) = c + 2 w p g(v) 2 dv + 2G(p )g(p ) + (1 G(w ))g(w ). (2) where we note that p is an increasing function of c. For large enough s, however, p > w and no consumer will choose to search beyond the first firm. Each firm is then a single-good monopolist that faces demand 1 G(p). It is easy to see that the resulting profit (p c)(1 G(p)) is maximized at the price p m that solves p m (c) = c + 1 G(pm ) g(p m. (3) ) The threshold search cost level s is such that w = p m at which point (3) and (2) coincide. Formally, s solves v p m (v p m )g(v) dv = s. (4) So for s s, the equilibrium price solves (2), for s > s the equilibrium price equals to the monopoly price p m (c). Figure 1 shows that the equilibrium price is continuous in search cost s and strictly increasing for small s and constant for s > s. Note that after s crosses the threshold value s equilibrium demand decreases discretely. For large s > s, consumers only buy when their match value at the first firm is large enough so that the market size equals 1 G(p m ). For smaller s, consumers will search on and total demand equals (1 G(p)) (1 + G(p)). As equilibrium price is continuous in search cost, the market size (and thus firms profits) in equilibrium is discontinuous at s = s. Figure 1 also displays this discontinuous jump in market size. 8

9 *** INSERT FIGURE 1 HERE **** On may wonder why retailers cannot prevent the market size from decreasing discretely by abstaining from increasing their prices above w ( s) (the reservation utility corresponding to s)? The answer is the following. Once s > s, even though both firms would prefer to collectively charge p 1 = p 2 = w (s), individually they have an incentive to increase their price to a level above w (s). This happens because once consumers believe prices do not exceed w they search, and thus firms have the incentive to increase prices to the levels given by 2, but at such prices consumers should not search, and thus in equilibrium both firms charge p m (c) and no consumer searches for s > s. Note that this explanation for the discrete fall in equilibrium market demand is related to the Diamond paradox (Diamond, 1971). ELABORATE As is known in the search literature, (REFERENCE?) note) also that when the first search is costly, instead of having a discrete fall in market size, the market would break down completely for high search cost. The reason is that when consumers expect prices to be larger than their reservation utilities, they will not search in the first place if the first search is costly. Symmetric Beliefs In the previous sections we have argued that consumers could also have symmetric outof-equilibrium beliefs, i.e., if consumers that have visited firm 1 observe it has deviated to p 1 p they expect firm 2 has deviated to the same price. We now proceed to constructing a symmetric equilibrium p(c) under symmetric beliefs, where to make the transition to the next section easier we already take into account that the equilibrium price may depend on cost (which is here known to consumers). For low enough s, so that p(c) < w, expected demand for firm 1 which charges p 1, while firm 2 charges the equilibrium price p(c) and all consumers expect the firm they have not visited first to charge the price they observed at the firm they just visited is given by: + w Q 1 (p 1 )G(w p(c) + p 1 ) + w p(c)+p 1 p 1 G( p(c) p 1 + v)g(v)dv p 1 G( p(c) p 1 + v)g(v)dv. (5) by The first order condition for profit maximization given the demand in (5), is then given 1 G( p(c)) 2 p(c) = c + 2 w p g(v) 2 dv + 2G( p(c))g( p(c)), (6) Recall that (6) is derived under the assumption p(c) < w. As w is decreasing in s, for large enough s we will have p(c) = w we have p(c) = w = c + 1 G( p(c))2 2g( p(c))g( p(c)). (7) It is not difficult to see that this price actually is the profit-maximizing price for a monopolist selling both products. Such a monopolist would set p 1 = p 2 = p (NEEDS TO BE PROVEN) and maximize (1 G(p) 2 )(p c), which is maximized at p jm that solves p jm (c) = c + 1 G(pjm ) 2 2g(p jm )G(p jm ). (8) This price is always higher than the single-product monopoly price p m because the joint profit maximization takes into account that some consumers who do not buy product 1 9

10 will buy product 2. Thus for some value of the search cost, the two firms set prices that are higher than the single-good monopoly price and as high as the price of a multi-good monopolist! So the largest search cost at which our solution in (6) still holds is where w (s) = p jm. Denote by s the search cost that solves v p jm (v p jm )g(v) dv = s. (9) Then for s s, equilibrium price is given by (6). As p jm > p m, it is clear that s < s. What about s > s? First note that in this case w < p(c). This means that whatever the symmetric equilibrium price, it cannot be smaller than w because in that case each firm wants to deviate to a larger price (as the symmetric equilibrium price is still given by p(c)), so they will at least charge w if consumers still search. Can it be the case that firms charge equilibrium prices above w? In this case consumers do not search and firm 1 s demand is the monopoly demand 1 G(p). For this demand profit is maximized at p m. Now depending on whether w is above or below p m we will have two cases. First, consider when w p m which happens when s < s < s. In this case firms will never want to charge any price above w, because they have an incentive to charge at most w. So the only possible candidate for equilibrium is p 1 = p 2 = w. We know that deviations to higher prices, where the profit function equals the monopoly profit function, are not profitable. Deviations to lower prices require to look at the quantity sold by the deviating firm. This quantity is given by (5) since both prices are at most w. In this case we know that profit is increasing in p 1 because the symmetric solution to the FOC is p that is larger than w. Thus, in this case p(c) = w and as w is decreasing in s, we have that the equilibrium price is decreasing in search cost. Now consider w < p m, which happens when s > s. Here for the same reasons as explained in the Wolinsky model, even though firms suffer a discrete drop in demand,they can no longer stop from deviating upwards to prices larger than w when consumers do search, and thus consumers stop searching and the equilibrium price becomes p m. Thus we have the following proposition. Proposition 4 Under symmetric beliefs, the Wolinsky model has a unique equilibrium where the retailers price p(c) is implicitly given by (6) for s < s, is equal to w for s s s, and is equal to p m (c) for s < s. Figure 2 illustrates for a given c how the equilibrium price varies with search cost in the Wolinsky model under symmetric beliefs. fo small search cost prices first rise, until the joint monopoly price, and then fall in s until price equals the monopoly price, and is then constant for even larger values of s. Other explanations for price decreasing in s. EXPAND Note that the non-monotonicity of the equilibrium price in search cost, does not depend on whether or not the first search is costly. With costly first visits, the monopoly pricing part of the Figure would disappear as the market breaks down (as before), but the part of the equilibrium construction where s < s < s would not be affected as expecting these prices, consumers would still consider making a first search, and given that consumers search, firms do not have an incentive to charge higher prices. 4 Asymmetric Information about common cost In this Section we show how symmetric beliefs naturally arise in a Wolinsky type of environment where firms have a common cost that is known to the firms, but unknown 10

11 p w p jm (c) p(c) p m (c) p (c) p d (c) s s s Figure 1: Equilibrium prices in Wolinsky (red) and random cost (blue) models for equal c. Dashed line depicts w while p d is the oligopoly price when consumers know both prices and their valuations for both products. (and uncertain) to consumers. the effect of common cost uncertainty has been studied in the context of the Stahl (1989) model (see, e.g., Dana (1994), Tappata (2009) and Janssen et al. (2011)), but has not been incorporated in the Wolinsky model. The timing of the model and the equilibrium concept we use has been described in Section 2. Proposition 5 Let p(c) and p m (c) be given as in (6), respectively (3). For any given s, the equilibrium of the asymmetric information model can be characterized as follows. If Definition 6 1. (a) p(c) < w (s), then the equilibrium is fully separating and the equilibrium prices are given by (6); (b) max(p m (c), p(c)) < w (s) < p(c), then the equilibrium is partially separating with equilibrium prices being given by (6) for values of c such that w (s) > p(c), while for values of c such that w (s) p(c) equilibrium prices are indepndent of c and equal to w (s); (c) p(c) < w (s) < p m (c), (d) p m (c) < w (s) < p(c) Since w is independent of c, for a range of c and sufficiently large s, equilibrium price will be independent of c and thus there will be partial, or full, pooling in equilibrium. Note that s and s depend on c, so whether conditions in Proposition?? are satisfied for some cost types depends on the range of costs and on s. Namely, if s s( c) we have a fully separating equilibrium where prices are given by (6). If s(c) < s < s( c) then all cost types charge a pooling price w so we have a fully pooling equilibrium. Several other partially separating equilibria are possible for other coefficients. 5 Retail cost is strategically set by an upstream firm We now turn to the environment where the retailers cost is determined by an upstream firm and consider the effect of asymmetric information about the retailers cost on the decision of the upstream firm and upstream and downstream profits. So we will consider 11

12 two modifications of the Wolinsky model. In the first one, an upstream firm will choose its price observed by consumers and sell the good to downstream firms. Given the upstream price, downstream firms will choose their prices. In the second version everything will be the same but consumers will not observe the upstream price. 5.1 Consumers observe the upstream decision on c For any c that the upstream firm may charge, from Section 3 we know what happens downstream. This is because when consumers observed the choice upstream, any price deviation by downstream firms can only be interpreted by that firm s individual deviation. This leads to belief structure in the Wolinsky model. Namely, for search cost and upstream price such that s < s(c) we have that prices are given by (2). For all other case the downstream price is p m (c). Given this, the upstream firm needs to set an optimal price for itself. One important consideration here is that the downstream demand, and hence the upstream demand is discontinuous at c such that s(c) = s. For c below this threshold upstream demand is given by 1 G(p) 2 while for c above this threshold demand falls to 1 G(p) because consumer stop search. Given that the upstream firm can control when s becomes larger than s, and given that the demand falls there discretely, we will have a situation where the upstream firm may accommodate larger search cost by reducing its price in such a way as to prevent downstream prices becoming larger than w. To formally solve the model, let us start with the case where s is relatively small, so the consideration from above does not apply. Then both retailers react to a choice of c by setting p (c) as in Wolinsky model. The upstream firm s demand consists of the demand at both retailers, and consumers buy at one of the two retail firms if, and only if, max(v 1, v 2 ) > p (c), and thus buy with probability 1 G(p (c)) 2. The upstream firm maximises and the optimal c o is given by: (1 G(p (c)) 2 )c, c o = 1 G(p ) 2 1 2G(p )g(p ) p c Here the derivative p c (co ) can be found implicitly from (2). This is true for such s that the resulting p (c o ) < w. Now let us look at such s that for the upstream price that solves (10) we have p (c o ) > w. In this case consumer would not search, and thus (10) is not the correct expression. In fact, downstream firms would revert to pricing at the monopoly price (corresponding to c) and upstream profits would suffer a discrete decline because consumers search less and buy less. Thus the upstream firm would not set such a price that consumers would stop searching immediately after s becomes large enough. This means that the optimal upstream price has to be such that p m (c) = w. In this case consumers (just barely) prefer to search, and downstream firms charge p = w. Since as s increases, w falls, this accommodation of downstream search by the upstream firm cannot continue forever. At some search cost, it will become profitable to let consumers stop searching, and charge the optimal price given that downstream firms charge a monopoly price for that c. Thus, from that moment on s drops out of the model and we are in the world on classical double marginalisation prices. Formally, let us define s o as search cost such that at c o that solves (10) we have p (c o ) = w (s o ). Also, let c m denote the upstream price that maximises (this is a classic double marginalisation price) (10) (1 G(p m (c)))c. (11) 12

13 c m is given by c m = 1 G(pm (c m )) g(p m (c m. (12) )) Finally, let s o denote a search cost such that (1 G(w ( s o )) 2 )c = (1 G(p m (c m )))c m where c is such that downstream firms charge w. Charging such a c or charging c m is equally profitable at s o. Now we can state the equilibrium formally. Proposition 7 In case consumers observe the price set by the upstream firm, the equilibrium is given by: (i) upstream price is given by (10) and downstream prices are equal to p (c) if s < s o ; (ii) upstream price solves p m (c) = w and downstream prices are equal to w for s o s s o ; (iii) upstream price is c m and downstream prices are p m (c) for s o < s The following figure illustrates equilibrium. Initially prices are increasing in s, but once s reaches s o prices start to decline, before they jump up and become flat at s > s o. p, c s o s o s Figure 2: Upstream (solid) and downstream (dashed) prices for the model with vertical relations and observed c for G( ) N(100, 15). 5.2 Consumers do not observe the upstream decision on c Now we turn to the more natural case when c is not observed by consumers. What changes when c is not observed? Most importantly consumers cannot change their beliefs p e i in response to change of c as they did with observed c. Also, when observing out-of-equilibrium prices, consumers now can blame either the retailer or the wholesaler for deviation. So the downstream reaction to deviations can be anywhere between the Wolinsky model where consumer think the retailer has deviated, to the model with random costs where deviations are always attributed to the marginal cost, thus wholesaler in this model, and so consumers think that the other retailer is charing the same price as the deviating one. There are possibilities in between where consumer attach intermediate probabilities to the retailer deviating, and wholesaler deviating, but for now we will only consider these two extreme cases. In this section we will concentrate on the model where upon observing non-equilibrium price, consumers always blame the upstream firm and thus beliefs are exactly as in the random cost model. 13

14 The advantage of these beliefs is that, once they are set, actual upstream demand does not depend on what consumers think equilibrium is. Instead, since they always set their beliefs in accordance to the firms price they observe, upstream demand only depends on actual prices and not on what consumers think the upstream firm s equilibrium choice was. This is in contrast with Janssen and Shelegia (2012) where the fact that consumer search according to what they think the equilibrium upstream price is drives the result. In our model the same issues would arise if consumer beliefs in the unobserved case were as in the Wolinsky model, thus it would matter what consumers think the upstream price was, and thus what they think prices elsewhere are when visiting the first shop. Unobserved c model with flexible beliefs Let us solve the unobserved wholesale price model for the beliefs where p e j = p i instead of p e j = ˆp as in the observed case. In this case, consumers upon observing a non-equilibrium price think that the same price will be charged by the other retailer, or equivalently they they think that retailers have not deviated from their equilibrium strategy, and it the upstream firm that has changed its price. Once consumers have these beliefs, we know that for a given c the equilibrium is as described by Proposition??. Note that now what consumers belief about c becomes irrelevant because consumer behavior only depends on actual downstream prices and does not depend on p u, their belief about equilibrium prices. It is again the case that the upstream firm will choose c to maximize its profits given that it expects downstream firms to behave as described in Proposition??. One difference to the previous model is that now for a range c downstream prices are independent of c. This happens when downstream firms charge w. This means that the upstream firm will always have an incentive to increase c in this range. So upstream firm will never charge c such that s (s(c), s(c)). To such a c, upstream firm will always prefer to charge c that solves s(c) = s. So either the upstream firm will set c that solves c u = 1 G( p) 2 1 2G( p)g( p) p c where p is defined in (6), or it will charge c u that solves (13) p m (c u ) = w, (14) or it will charge c u = c m. The last case will result in the traditional double-marginalization problem. Whether one of these upstream prices is feasible (if search costs are low, it may be impossible to choose c such that the upstream firms charge w or p m (c)) depends on the search cost parameter. When the search cost is sufficiently close to 0, it will never be profitable to charge c that solves (14) or c = c m and so the optimal upstream price will solve (13). As s increases, so does the profit from charging the solution to (14), and eventually the upstream firm will switch to this price. This leads to a discontinuity in terms of s. For small s the upstream firm will solve (6), but after some threshold is will jump to c that solves s(c) = s. There will be a jump because downstream prices are independent of c for a range of c, so once the switch happens, it happens to the upper limit of this interval. Eventually s is so large that the upstream firm cannot or does not want to induce downstream firms to charge w, and switches to charging the traditional double marginalisation price c m. In this case consumers do not search, so there is a drop in upstream demand. This implies that there will be another discontinuity is s, because a drop in 14

15 demand has to be compensated by a discrete increase in price. See the following figure for illustration. p, c s u s u s Figure 3: Upstream (solid) and downstream (dashed) prices for the model with vertical relations and unobserved c for G( ) N(100, 15). To state our results formally, let us define s u as the search cost at which the profit of the upstream firm (1 G( p(c)) 2 )c at the solution to (13) equals (1 G(w (s u )) 2 )c where c solves (14). If such s does not exist, let s u = 0. Also, define s u to be such a search cost that the upstream profit (1 G( p(c)) 2 )c at the solution to (14) equals the upstream profit at (1 G(ˆp(c m )))c m. (note that becuase of their identical definitions, s u = s o ). Again, if s u s u cannot be found, set s u = s u. We have the following proposition Proposition 8 In the unobserved c model with flexible beliefs there are the following equilibria depending on s: (i) If s s u then c u solves (13), downstream firms charge prices according to (6) and consumers search whenever v w ; (ii) if s u < s s u then c u solves (14), downstream firms charge p = w and consumers search whenever v w ; (iii) if s > s u, then c u = c m, downstream firms charge p m (c) and consumers do not search. Several important observations stem from Proposition 8. For small s (s < s u ), downstream prices are increasing in s but upstream prices may actually be decreasing in s (see Figure 4 for illustration). This is intuitive given that with larger s downstream margins increase, but the upstream firm may accommodate this by lovering its price so that even though downstream prices increase, upstream price falls. In the range of search cost [s u, s u ], downstream and upstream prices are decreasing in search cost. This is because ˆp = w and w is decreasing in s. Moreover, in this range also the upstream price is decreasing in s. This is because c u solves (14), and because w is decreasing in s, and p m (c) is increasing in c, so c u is decreasing in s. This range is interesting because comparative statics is reversed for a standard search model. Even though Janssen and Shelegia (2012) also show such relationship between downstream prices and search cost, the reasons are quite different. In this model the reason is that for large enough search costs downstream firms are forced to charge the reservation prices without reacting to their marginal cost. As w is monotonically decreasing in s, so ends up the retail price. In Janssen and Shelegia (2012) the effect comes from the fact that as downstream margins increase, the upstream firms incentive to increase its price shrinks, and the first effect may be stronger than the effect through higher margins. In Janssen and Shelegia (2012) the decreasing prices were shown for linear demand, but may have been 15

16 increasing for other demands, whereas here the result does not depend on the distribution of consumer tastes. Prices eventually reach the classic double-marginalisation level, after which they become independent of s. As one can see from the estimations in Figure 4, the range of search costs where prices are decreasing in s may be very large, and price swings depending on s may be substantial (about 15% in Figure 4). Finally, let us compare the model to models with observed and unobserved c. For s > s o the two models coincide. This is because once in the observed model the upstream firm switches to such c that downstream prices equal w, consumer beliefs become irrelevant for the determination of equilibrium price, and thus the two models are identical. For smaller search costs,(s u < s < s o ) in the observed case prices are still below w, but in the unobserved case they have already reached w. Before this happens (s < s u ), prices are lower than w in both models, but becuase in the unobserved case retail prices are higher for a given c (see the Random Cost model vs. Wolinsky model), overall prices are higher in the unobserved case, even if upstream prices are lower in the unobserved case. The latter is dues to the well-known fact that the larger the size of the downstream margin, the lower the upstream price but the higher the retail price. All these are illustrated below. p, c s u s o s u = s o s Figure 4: Upstream (solid) and downstream (dashed) prices for the model with vertical relations. Red curves are for observed c and blue ones are for unobserved c. In both cases G( ) N(100, 15). 16